What is the Fourier series for $\{a\}\{b\}$, i.e. the product of the fractional parts of $a$ and $b$. I know what the Fourier series looks like for a single value of either $a$ or $b$, but I want to know what it is when the two are multiplied together.
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1I'm not quite sure what a Fourier series for a function of two variables looks like, but what happens if you just multiply the two individual Fourier series together? – Gerry Myerson Dec 03 '12 at 06:00
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I get a complicated double series, which im not sure how to simplify. – Ethan Splaver Dec 04 '12 at 00:57
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What kind of simplification are you looking for? and do you have any reason to think the kind of simplification you are looking for actually exists? – Gerry Myerson Dec 04 '12 at 01:58
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If $0 \leq \{a\} < 1$ and $0 \leq \{b\} < 1$, then $0 \leq \{a\}\{b\} < 1$. So the Fourier series of the product would look just like the Fourier series of $\{c\}$ of some real number $c$.
glebovg
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But I need it to match up at the discontinuites of both a, and b – Ethan Splaver Dec 03 '12 at 05:21
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I think ${a}$ and ${b}$ are meant as real functions in the variables $a$ and $b$ respectively, not fixed numbers. – WimC Dec 03 '12 at 06:02
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@WimC Just define ${c} = {a}{b}$, which is the fractional part of some real number $c$. – glebovg Dec 03 '12 at 17:47